The large cells in early vertebrate embryos are organized by radial arrays of microtubules called asters. Asters grow, interact, and move to precisely position the cleavage planes of for each cell division. Cell-spanning dimensions are presumably required for interphase asters to explore the size and shape of the large cytoplasm. It has been unclear whether asters grow to fill the enormous egg according to the standard model of aster growth proposed in smaller somatic cells, or whether special mechanisms are required. In this dissertation, I combine biochemical reconstitution and biophysical modeling to propose a new model of aster growth that involves autocatalytic microtubule nucleation. By imaging asters in a cell-free system derived from frog eggs, I measure the number and positions of microtubules over time and find that most microtubules were nucleated away from the centrosome. I also find the interphase egg cytoplasm supports spontaneous nucleation after a time lag. Given these observations, I construct a biophysical model that describes aster growth from the interplay of microtubule polymerization dynamics and autocatalytic nucleation. This leads to the concept of a critical nucleation rate, which defines the quantitative conditions that predicts either (i) a growing aster characterized by a linear increase radius without dilution of microtubule density at the periphery, or (ii) a steady-state aster with small, constant radius. By combining theory and experiments, I propose a scenario where unbounded aster growth consists of individual microtubules that are themselves bounded in length. This offers a mechanistic explanation to how cells might differentially regulate aster size during the cell cycle. In summary, aster growth is a collective phenomenon of microtubules providing us with insight to how cells self-organize.